We have some historical remarks about Leonard pairs.
The concept of a Leonard pair originated in Algebraic Combinatorics,
in the study of \(Q\) -polynomial distance-regular graphs [1]}, [2]}, [3]}.
The origin story begins with the 1973 thesis of Philippe Delsarte [4]}.
In that thesis, Delsarte showed that a \(Q\) -polynomial distance-regular graph
yields two sequences of orthogonal polynomials that are related by what
is now called Askey-Wilson duality [5]}.
Motivated by Delsarte's thesis and Eiichi Bannai's lectures at Ohio State University,
Douglas Leonard showed in 1982 that the \(q\) -Racah polynomials give the most general
orthogonal polynomial system that satisfies Askey–Wilson duality [6]}.
In their 1984 book [1]} Bannai and Ito give a comprehensive version
Leonard's theorem, that treats all the limiting cases.
This version gives a complete classification of the orthogonal polynomial systems
that satisfy Askey–Wilson duality.
It shows that the orthogonal polynomial systems that satisfy Askey–Wilson duality all come
from the terminating branch of the
Askey scheme (see [8]} and [9]}.)
The Leonard theorem [1]} is a bit complicated.
To simplify and clarify the theorem, the second author introduced the
notion of a Leonard pair and Leonard system [11]}.
The Leonard systems are classified up to isomorphism in [11]}.
This result gives a linear algebraic version of Leonard's theorem.
For more information on Leonard pairs and orthogonal polynomials, see
[11]} and [5]}, [15]}.
